Exponential function: Difference between revisions

Revision as of 22:28, 4 March 2025

Revision as of 22:34, 4 March 2025

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<math display="block">e^{it} =\cos(t)+i\sin(t). </math>

This formula provides the decomposition of complex exponential ininto [[real and imaginary parts]]:

<math display="block">e^{x+iy} = e^x\,\cos y + i e^x\,\sin y.</math>

The trigonometric functions can be expressed in terms of complex exponentialexponentials:

<math display="block">\begin{align}

\cos x &= \frac{e^{ix}+e^{-ix}}2\\

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\end{align}</math>

In previousthese formulas, {{tmath|x, y, t}} aeare commonly interpreted as real variables, but the formulas remain valid if the variables are interpreted as complex variables. These formulas may be used forto definingdefine trigonometric functions of a complex variable.<ref name="Apostol_1974"/>

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